If I make the url for a zip file the href of a link and click the link, my zip file gets downloaded and opening it gets the contents as I expect.
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You're passing the URL to readfile() like:
$archive = 'https://mysite.com/uploads/my-archive.zip';
While you should pass the path on the server, for example:
$archive = '/uploads/my-archive.zip';
Assuming the file is located in the upload folder.
Additionally try the following headers:
header("Content-type: application/octet-stream");
header("Content-disposition: attachment; filename=file.zip");